Modern Instrumentation, 2012, 1, 1-7 Published Online January 2012 (
A Hybrid Angular/Energy Dispersive Method to Improve
Some Characteristics of Laboratory X-Ray Diffraction
Valerio Rossi Albertini1, Daniele Bailo1,2, Amanda Generosi1*, Barbara Paci1
1Istituto di Struttura della Materia del CNR, Roma, Italy
2Istituto Nazionale di Geofisica e Vulcanologia, Via di Vigna Murata, Roma, Italy
Email: *
Received December 30, 2011; revised January 20, 2012; accepted January 30, 2012
The Energy Dispersive X-ray Diffraction, generally referred as EDXD, has shown to be a valid alternative to the con-
ventional Angular Dispersive X-ray Diffraction, the ADXD. EDXD exhibits several advantages to its AD counterpart,
mainly related to the properties of the polychromatic X-ray beam utilized for diffracting, such as higher signal intensi-
ties, a wider accessible region of the reciprocal space, a greater transparency of samples, and a parallel data collection
of the q-points in the diffraction pattern acquisition. However, the main drawback of poly-chromaticity lays in the fact
that the quantities that modulate the scattered intensity in a diffraction measurement depend on the energy. These quan-
tities are the primary X-ray beam spectrum, polarization, and X-ray absorption, the last producing by far the most criti-
cal effect because it rapidly changes as a function of energy. Therefore, a detailed knowledge of the energy dependence
of all these quantities is required in EDXD in order to process the data correctly and prevent systematic errors. The dif-
ficulty in handling the energy-dependent factors complicates the experimental procedure and may make the measure-
ments unreliable. In the present paper, a hybrid method between the ED and AD X-ray Diffraction is proposed to main-
tain the advantages of the polychromatic nature of the radiation utilized in EDXD, while preventing the problems pro-
duced by the energy-dependent quantities.
Keywords: X-Ray Diffraction; Energy Dispersive; Angular Dispersive
1. Introduction
An X-ray diffraction pattern is generally defined as the
profile of the diffracted intensity vs. the deflection angle.
However, in this case, the energy of the primary radiation
must be specified. If such energy is increased (decreased),
for instance by substituting the X-ray tube anode and se-
lecting another fluorescence line, the diffracted intensity
profile undergoes a stretching (compression). To plot a
“universal” diffraction pattern (i.e. a pattern that is de-
pendent on the sample only and, therefore, invariant to
changes in the primary X-ray beam energy), the angular
variable must be replaced with the quantity on which the
diffracted intensity actually depends, i.e. the momentum
transfer Δ
, whose amplitude takes the name “scattering
parameter”. The latter can be expressed in terms of the
radiation energy E and of the deflection angle 2
q = a·E·sin
where a is a constant equal to 1.014 Å–1/keV. With refe-
rence to the previous example, an increase (or decrease)
of the radiation energy will be counterbalanced (see Equa-
tion (1)) by a decrease (or increase) of the angle, so that
the pattern profile plotted as a function of q will appear
unchanged for any choice of the primary beam energy.
An important consequence of the above equation is that
two ways are actually available to perform the q-scan in
order to collect a diffraction pattern [1,2].
The first approach, namely the Angular Dispersive me-
thod, is the standard one. It consists of carrying out an
angular scan by making use of a monochromatized X-ray
beam, usually a fluorescence line of an X-ray tube anode.
Conversely, the q-scan can be performed through an
energy scan of the diffracted beam at a fixed angle. In
this approach, called the Energy Dispersive mode, a con-
tinuous energy spectrum radiation is used as the primary
beam, while the scan is performed electronically, rather than
mechanically, by a solid state detector [3]. The signal pro-
duced by the detector is amplified and the Analogical/
Digital conversion is accomplished and visualised by a
multichannel analyser, in the form of a frequency histogram.
In the histogram, each channel corresponds to a small
energy interval. The detected photons are piled up in the
channels corresponding to their energy, thus allowing the
digital reconstruction of the radiation energy spectrum.
The main advantage of the ED mode comes from the
*Corresponding author.
opyright © 2012 SciRes. MI
high energy components of the white X-ray beam [4]. This
advantage appears evident when considering the differ-
rence between the production mechanism of the fluores-
cence lines used as primary beam in the ADXD and that
of the Bremmsstrahlung (breaking radiation, primary beam
in the EDXD): while the fluorescence lines are emitted at
well defined energies, the Bremmsstrahlung energy spec-
trum ranges from zero to a maximum value dependent on
the voltage of the X-ray tube power supply used. A po-
wer supply set at n kV will produce the emission of a
Bremmsstrahlung up to n keV.
Among the X-ray tubes commercially available, supp-
lied with standard high power generators, the Ag anode
emits the most energetic fluorescence line (kradiation
at about 22 keV). In other words, this is the highest en-
ergy limit for X-ray radiations usable as primary beams
in laboratory ADXD, see red line in Figure 1.
The use of a high energy primary beam prevents the
effects due to the low transparency of the materials at low
energies. Such low transparency is the principal reason for
weak diffracted signals when the samples contain heavy
elements (an example of the radiation transmitted by a 10
μm thick Pt powder sample is shown in Figure 1(b)).
Although some difficulties can be circumvented by
thinning the sample or using reflection geometry, rather
than transmission geometry, X-ray absorption still repre-
sents a major obstacle to accomplishing fast and high sta-
tistics measurements.
To quantify the advantage of using high energy radia-
tion, it is worth stressing that the X-ray intensity reduce-
tion due to absorption A (Figure 1(c)) is exponentially de-
pendent on the mass absorption coefficient μ/ρ Lambert-
Beer law)
A=exp μρρx
[5]. The radio μ/ρ, in turn,
is substantially proportional to λ4. As a consequence, the
decrease in the transmitted intensity across a sample of
given density, thickness and chemical composition, roug-
hly scales as exp (–λ4). An increase of the wavelength may,
therefore, result in a dramatic drop of the detected signal.
Another advantage of ED to AD can be illustrated
through Equation (1). The width of the reciprocal space
window accessible in a diffraction experiment is
MAX MAXminmin min
qqq, where q
a Esinandqa Esin
 
In the ADXD, the energy being fixed, the amplitude of
the interval only depends on the difference between
the lowest and the highest angles reached during the an-
gular scan.
However, in the EDXD, the further degree of freedom
represented by the variable energy allows for the explo-
ration of a much wider q-range. Indeed, assuming in both
cases a negligibly small min
, and an upper limit MAX
equal to 30˚.
In AD, for the reference Cu Kα fluorescence line,
AD 4 Å–1 , and about three time this value, for Ag
Kα in ED, in typical working conditions of the X-ray tube
power supply (HV = 60 keV), a white beam has suffi-
cient intensity to perform diffraction measurements in the
(10 - 50) keV energy range (see Figure 1). Therefore, a
even higher than 20 Å–1 can be reached, that is to
say, a value usually attainable by using neutron diffract-
tion large scale facilities.
In the transmission geometry (more reliable than refle-
ction geometry because the optical path in the sample is
independent on the photon energy), the EDXD master equa-
tion connects the intensity observed at the detector Iobs
with the other quantities involved in an ED X-ray measu-
rement [6]
 
obs0 0
AE, IE,E',
 
K is a factor that expresses the ratio between the total
scattered radiation (both elastically and inelastically) Iscatt
and the intensity scattered by a single stoichiometric unit.
In transmission geometry, it scales as 1/cos
P (E,
) = P0 (
) +
(E) (sin22
/2) is the polariza-
tion factor, where P0 = (1+cos2(2
))/2 corresponds to
the polarization due to diffraction of an initially unpolari-
zed primary beam, while
is the actual polarization of the
primary beam;
A(E, )A(E,0)
 is the energy dependent X-
Ray absorption factor in transmission geometry and A (E,
0) is the absorption coefficient defined in § 1;
E-E is the Compton shift.
Figure 1. Primary X-ray beam spectrum (a); transmitted
intensities through the Platinum powder samples (b); ab-
sorption coefficient (c) as calculated from the ratio between
the previous two curves.
Copyright © 2012 SciRes. MI
2. The Hybrid ED/AD X-Ray
Diffraction Method
The basic idea behind the method we propose is to take
advantage of the merits of both the EDXD and the AD-
XD by carrying out an ADXD-like measurements using
an EDXD diffractometer. Therefore, from the operative
point of view, the measurement is similar to the ordinary
ADXD one, consisting of the collection of the diffracted
intensity, from a minimum (0
) to a maximum (n
) an-
gular value, at steps of k
= 0
+ k
stant increment). The difference is that instead of an or-
dinary monochromatic primary beam, like in a real AD-
XD measurement, a polychromatic beam is used and the
diffracted radiation is collected by an energy dispersive
detector. As a consequence, at each k
, a whole diffra-
cttion pattern is acquired, instead of a single q-point as in
the ADXD.
A first merit of the hybrid method we propose is that,
unlike the ADXD measurements, in which monochroma-
tization imposes the suppression of all but the selected
energy component, all the components are preserved and
are collected in parallel, channel by channel (as in ED-
XD). Therefore, the same technique adopted to draw an
ADXD diffraction pattern, namely recording the intensity
as a function of k
, can be used here by plotting the in-
tensity contained in each channel.
To express this concept in formulas, we may say that,
given a generic channel j (corresponding to an energy Ej),
the diffraction pattern obtainable by plotting the number
of photons contained in j as a function of k
, is the cu-
rve I vs. (Ej, k
), or rather, according to Equation (1), I
(Ej·sin k
). Since all the channels can be used for this cal-
culation, a number of diffraction patterns equal to the nu-
mber of channels is obtained. Of course, for each choice
of the energy Ej, the amplitude of the q-interval increases
with k
and its extremes shift towards higher values along
the q axes. Indeed, the extremes of the q-interval can be
expressed as:
kminmink kMaxMax
qa Esinandqa Esink
 (3)
and, therefore, and , as well as the q-interval
( =–), increase with k
Such a procedure can be simplified to a certain extent,
considering that, due to the finite energy resolution of the
detector, adjacent channels are not independent. Thus,
instead of analysing the intensity of every single channel,
adjacent channels can be grouped and their collective
intensity can be analysed altogether. Since the energy
interval in our setup is 50 eV, while the energy resolution
of the detector is about 200 eV, the channels can be
grouped 5 by 5.
All the diffraction patterns obtained in this way can be
finally utilized (taking in account their relative q-shift) to
compose an overall pattern that contains all the informa-
tion present in each single pattern, as will be shown in
the experimental section.
3. Hybrid ED/AD Measurements and
Problems Connected to Data
Correction and Refinement
In Figures 2(a) and 2(b) the sequences of ED diffraction
patterns collected as a function of k
are shown in the
form of 3D-maps.
The maps refer to an Al and a Pt sample, respectively.
They clearly exhibit the characteristics of diffraction in
either ED or AD mode. The Bragg reflections appear in
the (Ej, k
) plane as “crests”, and progressively shift to-
wards lower energy values as k
is increased (accord-
ing to Equation (1), q = a·Ej·k
), following iso-q cu-
The crests intensity is modulated by the occurrence of
phenomena concomitant to diffraction in an X-ray mea-
surement, as will be discussed later. Vertical sections of
the crests with planes parallel to the E axis are the energy
dispersive patterns collected in correspondence of the
-value at which the cutting plane is placed.
Figure 2. Sequences of patterns (a) Aluminium; (b) Plati-
num) collected as a function of energy and angle. The re-
sulting crests represent the traces of Bragg reflections in
their path along constant q-curves (approximately hyper-
bolic) due to the angular scan.
Copyright © 2012 SciRes. MI
Conversely, vertical sections of the crests with planes
parallel to the
axis at a certain E-value, are equiva-
lent to conventional AD patterns collected in correspond-
dence of that E-value. Two planes of this kind at Ej = 40
keV and k
= 15˚, respectively, are evidenced in Fi-
gure 2(b) in red. First of all, we can make some general
observations concerning these maps. Due to the low elec-
tron number, the crests in Figure 2(a) span from low to
medium q-values, being absent in the high q-zone becau-
se of the rapid decrease of Al coherent form factor as a
function of q. In contrast, in Figure 2(b), Pt crests are
present also in the high q-zone, but their left tails tend to
vanish as a consequence of the intense X-ray absorption
in the low energy region. In such a region, the absorption
gets even more intense with increasing angles, so that the
left tails diminish faster at higher k
Secondly, on the basis of the master Equation (2), we
can make a deeper preliminary inspection prior to data
refinement. Let us take, for instance, the most intense peaks
of Figure 2(a) and (b). Their intensities have maxima in
the central part of the crests. This behavior is dominated
by the three main factors influencing the height of Bragg
reflections in the ED mode, namely, by the energy-de-
pendent coefficients of Iscatt in Equation (2). At low ener-
gies, both the primary beam intensity and the sample trans-
parency are low, so that the peak amplitude lowers as well.
At high energies, the transparency is high, but the inten-
sity of the primary beam becomes vanishingly small, so
that the result is, again, the decrease of the peak ampli-
In standard EDXD, these effects have to be taken into
account and the data accurately corrected for them. Othe-
rwise, major systematic errors, which produce profile di-
stortions and make the measurement unreliable, occur
For this reason, much effort has been dedicated to co-
rrectly determine quantities that must be measured inde-
pendently of the diffraction pattern collection.
Unfortunately for the experimenter, such measurements
are far from trivial, as discussed below.
3.1. Primary X-Ray Beam Spectrum, I0 (E)
The determination of the primary beam spectrum is a pro-
blem that dates back to the invention of the EDXD tech-
nique [8]. The difficulty is due to the fact that a direct
measurement of I0 is impossible because its high intensity
“dazzles” the energy sensitive detector. Therefore, the tech-
nique of measurement must be to collect a signal propor-
tional to I0, but with a much lower intensity. Several ways
have been suggested to do this, the most common being 1)
an extreme collimation of the radiation; 2) the reduction
of the X-ray tube power supply current; 3) the measure-
ment of a rare gas scattering.
1) A high collimation reduces the beam cross section,
and a direct acquisition of the incident beam becomes po-
ssible by placing both the source and the detector arms in
horizontal position [8]. However, two problems arise. The
first is that, although the total photon flux is actually re-
duced, the power density of the so-obtained thin beam is
as high as in the original primary beam. This may produ-
ce non-linear effects in the detector diode, where a high
number of electron-hole couples are generated in a small
volume per unit time. Furthermore, the collimation slits
are partially transparent at high energies, so that the col-
limated beam contains an extra-contribution in its upper
part [9].
2) The reduction of the power supply current to the re-
gion of μA is also not perfectly reliable, since the focal
spot on the anode may suffer unpredictable changes when
the tube is forced to operate in such extreme conditions.
Therefore, although theoretically a reduction of the cur-
rent at a parity of high voltage should decrease the am-
plitude of the energy spectrum leaving the profile shape
unchanged, distortions must nevertheless be expected [9].
3) Finally, the measurement of the scattering from a rare
gas as Ar has been proposed [10]. This method is based
on the simplification of the master equation when Ar is
measured at small angles:
obs0 Ar
where N is the number of irradiated atoms and fAr is the
coherent form factor. Indeed, the Compton scattering, at
low angles and for a high atomic number sample, is neg-
ligible; the cross (interatomic) interference term in a mo-
noatomic gas is practically zero, as well as the absorption
(due to low density). Furthermore, also the polarization
disappears from the equation, because the measurement
is accomplished at a fixed angle and the energy-depen-
dent part is proportional to 2
sin 2
small (Equa-
tion (2)). Thus,
Nevertheless remarkable deviations from the theoretic-
cal energy spectrum profile have also been observed in
this case.
3.2. Polarization, P (E,
The second problem concerns the determination of the
primary beam polarization. It is well known that Brem-
msstrahlung is polarized because, especially at high ene-
rgies, the photons produced by the breaking of fast elec-
trons tend to be emitted in a preferential direction, that is,
in the direction of provenience of the electrons. In turn,
the polarization of the scattered radiation is composed by
two terms, P0 and
(E), (see Equation (2)). The initial
(E) of the primary beam is extremely di-
Copyright © 2012 SciRes. MI
fficult to determine, since it would require two accurate
measurements in two directions perpendicular to each
other and to the scattering plane [10]. Hanson et al. [11]
proposed to rotate the X-ray tube by 45˚, in order to coun-
terbalance the anisotropic emission of Bremmsstrahlung,
accepting the consequent geometrical complications, just
to prevent the polarization problems. In addition, calcula-
tions based on theoretical models are risky, since syste-
matic errors might be introduced in the final results due
to the difference between the model and the actual shape
of the anode focal spot (as well as of the emission me-
3.3. X-Ray Absorption A(E,
The absorption is by far the dominant effect on the am-
plitude of the EDXD patterns and produces dramatic
distortions, especially in the low energy part.
The usual way to measure it consists of collecting the
incident beam spectrum in the extreme collimation mode
described above and, then, leaving the setup unchanged,
repeating the measurement after the sample is placed along
the beam pathway. Normalizing the latter spectrum to the
former, the absorption coefficient in the direct configure-
tion (
= 0) is obtained (see Figure 1). Taking into ac-
count the change of the X-ray beam optical path in the
sample as a function of the angle, the absorption coeffi-
cient can be calculated at any angle.
However, in this case, along with those connected to
the measurement of the primary beam spectrum in the ex-
treme collimation mode, problems arise as a consequence
of the normalization procedure. In the low-medium ene-
rgy range, due to the strong absorption, the transmitted
beam may have very low intensity even after long collec-
tion times, so that the statistics is poor and errors on the
real form of the absorption are likely to occur.
4. Examples of Application of the
Hybrid Method
In the hybrid method we propose, the analysis is per-
formed considering each channel separately, i.e. at con-
stant energy. Hence, there is no need to determine the three
energy dependent coefficients previously described, as it
would be the case in an ADXD measurement. Indeed,
theoretically, if such coefficients were known at a high
degree of accuracy, no difficulty in the correct calcula-
tion of the overall diffraction pattern would occur.
This calculation could easily be done considering that
a map like those in Figure 2(a) and (b) represent the in-
tensity observed at the detector in correspondence to ea-
ch couple of values (Ej, k
). Once the three factors were
determined at (Ej, k
), as well as the incoherently scat-
tered intensity, the coherently scattered intensity could be
isolated. The latter is the quantity of interest and its con-
tribution to the overall diffraction pattern could be pro-
vided using the so-obtained value to improve the statis-
tics of the pattern intensity at j,kj k
 .
Unfortunately, for our discussion, the correction of the
observed intensity is imperfect and this method would
not be reliable. The same applies if the calculation of the
diffraction pattern is carried out by “slicing” the crests
along lines parallel to the
axis, or generally lines not
parallel to the E-axis, that is to say if the vectors lying on
these lines have a component along the E-axis (an exam-
ple of the this line type is the iso-q lines set mentioned
above). Again, the accurate knowledge of the three ene-
rgy-dependent factors would be required. Examples of
discrepancies due to uncertainties on the three factors can
be noticed by comparing the results reported in the lite-
rature concerning various EDXD studies carried out on
the same samples [6,7,12,13]. In our case, the discrepan-
cies can be shown (Figure 3) by observing the three Pt
diffraction patterns calculated a) from a theoretical mo-
del, b) by slicing the crests in Figure 2(b) at fixed-E (=45
keV), c) by slicing the crests at several
(8.0˚, 14.0˚,
16.0˚, 19.0˚, 22.0˚, 25.0˚) and, then, connecting the vari-
ous q-zones to cover the same q-range of pattern b). In
the theoretical calculation, the Caglioti parameters [14],
which take into account both the energetic and the angu-
lar contributions to the experimental q-resolution, were
obtained by accomplishing the best fit with the pattern
profile b). The agreement between a) and b) is evident,
while the relative intensities of the peaks in pattern c) are
rather incorrect. Therefore, if the slicing is carried out along
lines parallel to the
axis, a fundamental simplifica-
tion is obtained, which makes the measurements much more
Figure 3. Comparison among the Pt corrected and normal-
ized diffraction patterns calculated (a) the theoretical dif-
fraction pattern (black line), (b) by applying the ADXD
method to the channel corresponding to 45 keV (blue line)
and (c) by applying the EDXD method (multicoloured, to
evidence the 7 partial patterns acquired at selected angles
and suitably connected).
Copyright © 2012 SciRes. MI
axis is preferential from the point of view of
the energy-dependent coefficients, because in this direc-
tion the dependence on E ceases and the coefficients can
be easily calculated in the standard way utilized for AD-
XD data correction. In particular, the polarization and the
absorption terms will have the standard (geometric) de-
pendence on the angle only, as in ordinary ADXD, while
I0 is constant.
As a consequence of this consideration, the main obsta-
cle in EDXD, that is the fact that the data correction is
overly-complex due to the simultaneous presence of many
energy components, is completely overcome.
The patterns obtained by systematic slicing of the cre-
sts in Figure 2 at each Ej (corresponding to channel j or,
rather, to a group of adjacent channels) are shown in Fi-
gure 4 in the form of a 3D-map. They can be regarded as
portions of the overall pattern that can be reconstructed
by their summation.
The sum must be accomplished by taking into account
that the intensity at each q value of the overall pattern has
to be normalized to the number of partial patterns giving
a contribution at that q-value. Indeed, since the extremes
of a partial pattern are given by Equation (3), not all of them
contain the low and the high q-values. Therefore, the sta-
tistics at these values will be poorer than that at interme-
diate values. However, this effect is small and the overall
pattern exhibits a very good signal-to-noise ratio, as can
be seen in Figure 5. To highlight the gain obtained in the
statistics, a portion of the partial pattern is also shown in
the inset of the same figure (b) of Figure 3.
Figure 4. Diffraction patterns calculated at each energy, ac-
cording to the angular dispersive method. Notice that the
values of the extremes of each q-interval increase with in-
creasing E, as well as the q-interval amplitude.
Figure 5. Comparison among the patterns (a) and (b) al-
ready shown in Figure 3 (reported also here for the sake of
clarity), (c) the overall pattern calculated by applying the
proposed AD/ED hybr id mode, which demonstrates the im-
provement in statistical accuracy. A good agreement bet-
ween (a) and (c), up to q-values as high as 18 Å–1, can be
5. Conclusions
The hybrid method described in this paper overcomes
some of the limits of the two diffraction techniques ap-
plied separately. Indeed, on one hand, it involves simple
and reliable data processing, as in the ADXD mode and,
on the other hand, it takes advantage from the high flux
and high energy of the primary beam photons, as in the
EDXD mode. In particular, it is recommendable when
X-ray opaque samples are investigated, high-q reflections
have to be measured and an elevated signal-to-noise ratio
is required.
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